I have heard of this. It may be one of those things that is so crazy that it is actually true!

I have not heard of this, but I would like to know how it works

I have heard of a version of this. Years ago it was called "zig zag dieting." There is some weight loss lore that if you do the same thing day after day, your body will learn the pattern and then get stuck in a rut. When folks plateau and get stalled out, I've seen others advise them to increase calories a bit to fool the body into thinking something different is going on. Sounds like this is a version of not doing the same thing every day.

Firstly 2000 is not high calorie level, that is pretty much what you should eat on a average day if you want to stay at your current weight. So what he is talking about is eating at a safe weight loss level one day (1200) and then at a normal level the next (2000). That will of course work, as long as you stick to the 2000. It is all about the calorie deficient, so one day you are 800 under, and the next you are hopefully burning a bit extra from exercise, so on average you will probably have a deficient of 500 a day for the week. Thats 3500 calories, which is about 350 grams of fat burnt. So with exercise you are looking at a loss of about 0.5 kg a week, which is a great trend. And you wont be feeling like you never get to eat, since every second day you will be eating a little more, so less hungry, and probably a bit happier.
This is pretty much what i do. I aim every day for 1400, and every few days i treat myself with something, reaching about 2100. I exercise regularly, and i am losing about 0.5 kg a week. I am in for the long haul, so this really does work. Most days i don't even think about that i am on a diet, and the days when i treat myself, i still stay at a good calorie level. No mental, i failed again, every time a piece of chocolate passes my lips. I am trying to create a new lifestyle, not go on a trend diet.
Word of caution though, this will ONLY work if you stick to the upper limit, the 2000 calories, because if you eat 1400 and then the next day eat 3000 + you will GAIN weight. Your total calorie intake for the week is what counts, Not the one day low, one day high calorie idea.

That makes sense. I don't usually eat over 2000 on any given day anyway, but with having hypothyroidism I lose VERY slowly, so I guess I will give it a try.

There is nothing wrong with losing it slowly. You didn't gain it all in a month, so expecting to lose a significant portion in a month is just folly.